One key aspect of backgammon match play which is often confusing and kind of scary for new players is the use of ''Match Equity Tables, METs''. I don't pretend to be any good at using these but they are one aspect of backgammon which I have tried to start incorporating into my game. It should be noted that these equity tables are irrelevant for cash games and pretty useless if you don't know your equity of a particular situation.

These are basically charts based on rollouts + empirical data which show the probability of person A winning a match based on the distance of player A + B from the end. Below is the match equity used by Gnubg although you should be aware that the exact numbers are a matter of judgement and discussion.



OK, so on the vertical and horizontal bar you will notice ''1-away, 2-away etc'' these should be pretty explanatory, player A is on the horizontal and B is on the vertical. So for example, if the score is 11/5 in a match to 21, the odds of B winning the match are 26.5%, also, note how the diagaonal cells are each 50%, i.e. it assumes an equal skill level.

So once you know your odds of winning the match, you need to find the equity of this particular game such that you are indifferent between passing and accepting a double.

To start with we can use the simplest of examples:

Assume it is 5/5 at the start of a game (length is irrelevant), first you need to know your odds of winning the match - its 50%. So when you are offered a double by your opponent, you need to work out the inequality such that you work out the game winning threshold.

By accepting a double, you are risking an extra point (losing 2 points as opposed to 1 point) with the potential of gaining 3 points (winning 2 points compared with losing 1 point).

Now for some algebra: To work out this threshold, we need to make the equity for passing and accepting the same, where P is probability of A winning.

i.e. E = P*2-(1-P)*2
E = -1

Therefore -1 = 2P -2 + 2P

P = 25%


the minimum equity A needs to call a double is therefore 25%.

Generalised the equations can be written where G is amount gained after winning the double, M is the amount lost after losing the double and L is immediate loss from passing the double. In this example you will notice that G=2L and G=L although this will change with the use of the MET, it is the altering ration between G, M and L which makes equity tables useful and match doubling hazardous.

E = P*G - (1-P)*M for accepting
and
E = -L for passing


Therefore minimum equity in a given game:

P = (L + M)/(G + M) where L, M, G are the changes in match equity resulting from different game outcomes.



If we go back to our original example where A is winning a 21 point match 11/5, if B accepts and loses, his loss in equity is 26.5% to 18.8% = 7.7% In comparison, he can increase his equity from 26.5% to 33.1% = 6.6% by winning. By passing he loses equity of 26.5% to 22.6% = 3.9%

As a result his equation looks like -0.039 + P*0.066 - (1-P)*.077

P = .038/.143 = 26.6%

Therefore B should only accept if his odds of winning are >26.6%


This only scratches the surface of METs. The tables are too long to learn by heart so there are multiple estimation methods you can use which I wont try explaining here. They can become very complex very quickly once you start including gammons and backgammons and redoubles etc. There are hundreds of articles written online which probably explain this far better than me but this should hopefully get you started.

I'm sure i've messed up some of the maths and theory here so feel free to correct, comment, criticise, discredit etc.


Footnotes:

1) MET's are only relevant for matches
2) They become more useful the larger the spread between you and your opponent relative to the winning persons distance from the end - the larger the spread the bigger your errors are by sticking to the 25%-75% cash game cube threshold rule.
3) The closer you (as the losing player) are behind your opponent, the higher your accept threshold should be. As a counternote, the greater your advantage on your opponent, the more inclined you should be to double.

Nice work (-:

I've never really been able to do use the MET for decisions during the game but, then, I also seldom play matches.

Now, how is it about offering doubles as a trailing player? The intuition tells to me to wait for a bigger advantage than usual if I'm trailing but I'm really not sure.

I just thought about your 11-5 situation in a 21 point match. If I'm the guy who's trailing and think about doubling how much equity would I need before doubling.

By not doubling I can win 3.2% ME or lose 3.9% ME.
If I double I can win 4.6% ME or lose 11.3% ME.

So setting 3.2 * P - 3.9 * (1-P) = 4.6 * P - 11.3 * (1-P) and solving yields P = 84%.

Is my reasoning here correct? If yes, this means I would only get to double in races with a mega pip lead. In other positions than races I might be in a position where I play on to win a gammon?

I'm curious why you would find this intuitive. The opposite is usually correct and I must admit I find that intuitive as well. If you're trailing then you're happy to play for more points, because you already need a bit of luck to turn the match around. On the other hand the leading player should pass earlier and try to protect his lead especially if the position is gammonish.

So if you're trailing (a lot) in a match it can actually be correct to double in an almost even position if you have gammon chances if things go well.

I think you can compare it to being behind in a HU poker match. If you're opponent has you outchipped 3:1, would you tighten up or would you start pushing more marginal hands?

I can't really figure out how to use the OP's formula. The simplest way to calculate it imo, is: loss from doubling/(loss from doubling+gain from doubling)

(this is probably exactly what OP's formula does, I just feel this is easier to work with).

In your example you gain 3,4% by doubling if you win (from 29,7% to 33,1%, and you lose 3,8% by doubling if you lose (from 22,6% to 18,8%) so your doubling window would open at 3,8/(3,4+3,8)=52,8%.

So your doubling window actually opens a bit higher than 50% at this specific matchscore, but not by much. This is not that uncommon on an initial double, but on redoubles (or in very gammonish positions) it can often shift dramatically at lopsided scores.

Still keep in mind, that you shouldn't double just because you're in the doubling window. You should still factor in your opponents takepoint and the chance of losing your market on the next roll.

yes you are quite right about all this - I just wanted to show how it is derived in case anyone was interested. They exactly the same.

My line of thinking was that the opponent could take more loosely if he's in the lead, plus I was bit misled by my rubbish calculation.

But now if I come to think of it, this is the wrong direction. In a post crawford game the leader drops if he is below 50%, because he's 50% to win the next one and the match.

But to be honest I don't have much of a clue because I hardly play any matches at all. I'm more into money games and really didn't know for sure.

I'm also a bit confused about OP's footnote 3) that states that the leading player should double lighter as well. So both are inclined to raise the stakes early.

Like in the HU analogy: "I just put him all-in to get over with it"?